direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic10, C15⋊3Q8, C60.3C2, C20.1C6, C12.3D5, C6.13D10, Dic5.1C6, C30.13C22, C5⋊(C3×Q8), C4.(C3×D5), C2.3(C6×D5), C10.1(C2×C6), (C3×Dic5).2C2, SmallGroup(120,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic10
G = < a,b,c | a3=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic10
►Regular action on 120 points
Generators in S120
(1 29 48)(2 30 49)(3 31 50)(4 32 51)(5 33 52)(6 34 53)(7 35 54)(8 36 55)(9 37 56)(10 38 57)(11 39 58)(12 40 59)(13 21 60)(14 22 41)(15 23 42)(16 24 43)(17 25 44)(18 26 45)(19 27 46)(20 28 47)(61 105 87)(62 106 88)(63 107 89)(64 108 90)(65 109 91)(66 110 92)(67 111 93)(68 112 94)(69 113 95)(70 114 96)(71 115 97)(72 116 98)(73 117 99)(74 118 100)(75 119 81)(76 120 82)(77 101 83)(78 102 84)(79 103 85)(80 104 86)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 66 11 76)(2 65 12 75)(3 64 13 74)(4 63 14 73)(5 62 15 72)(6 61 16 71)(7 80 17 70)(8 79 18 69)(9 78 19 68)(10 77 20 67)(21 118 31 108)(22 117 32 107)(23 116 33 106)(24 115 34 105)(25 114 35 104)(26 113 36 103)(27 112 37 102)(28 111 38 101)(29 110 39 120)(30 109 40 119)(41 99 51 89)(42 98 52 88)(43 97 53 87)(44 96 54 86)(45 95 55 85)(46 94 56 84)(47 93 57 83)(48 92 58 82)(49 91 59 81)(50 90 60 100)
G:=sub<Sym(120)| (1,29,48)(2,30,49)(3,31,50)(4,32,51)(5,33,52)(6,34,53)(7,35,54)(8,36,55)(9,37,56)(10,38,57)(11,39,58)(12,40,59)(13,21,60)(14,22,41)(15,23,42)(16,24,43)(17,25,44)(18,26,45)(19,27,46)(20,28,47)(61,105,87)(62,106,88)(63,107,89)(64,108,90)(65,109,91)(66,110,92)(67,111,93)(68,112,94)(69,113,95)(70,114,96)(71,115,97)(72,116,98)(73,117,99)(74,118,100)(75,119,81)(76,120,82)(77,101,83)(78,102,84)(79,103,85)(80,104,86), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,66,11,76)(2,65,12,75)(3,64,13,74)(4,63,14,73)(5,62,15,72)(6,61,16,71)(7,80,17,70)(8,79,18,69)(9,78,19,68)(10,77,20,67)(21,118,31,108)(22,117,32,107)(23,116,33,106)(24,115,34,105)(25,114,35,104)(26,113,36,103)(27,112,37,102)(28,111,38,101)(29,110,39,120)(30,109,40,119)(41,99,51,89)(42,98,52,88)(43,97,53,87)(44,96,54,86)(45,95,55,85)(46,94,56,84)(47,93,57,83)(48,92,58,82)(49,91,59,81)(50,90,60,100)>;
G:=Group( (1,29,48)(2,30,49)(3,31,50)(4,32,51)(5,33,52)(6,34,53)(7,35,54)(8,36,55)(9,37,56)(10,38,57)(11,39,58)(12,40,59)(13,21,60)(14,22,41)(15,23,42)(16,24,43)(17,25,44)(18,26,45)(19,27,46)(20,28,47)(61,105,87)(62,106,88)(63,107,89)(64,108,90)(65,109,91)(66,110,92)(67,111,93)(68,112,94)(69,113,95)(70,114,96)(71,115,97)(72,116,98)(73,117,99)(74,118,100)(75,119,81)(76,120,82)(77,101,83)(78,102,84)(79,103,85)(80,104,86), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,66,11,76)(2,65,12,75)(3,64,13,74)(4,63,14,73)(5,62,15,72)(6,61,16,71)(7,80,17,70)(8,79,18,69)(9,78,19,68)(10,77,20,67)(21,118,31,108)(22,117,32,107)(23,116,33,106)(24,115,34,105)(25,114,35,104)(26,113,36,103)(27,112,37,102)(28,111,38,101)(29,110,39,120)(30,109,40,119)(41,99,51,89)(42,98,52,88)(43,97,53,87)(44,96,54,86)(45,95,55,85)(46,94,56,84)(47,93,57,83)(48,92,58,82)(49,91,59,81)(50,90,60,100) );
G=PermutationGroup([[(1,29,48),(2,30,49),(3,31,50),(4,32,51),(5,33,52),(6,34,53),(7,35,54),(8,36,55),(9,37,56),(10,38,57),(11,39,58),(12,40,59),(13,21,60),(14,22,41),(15,23,42),(16,24,43),(17,25,44),(18,26,45),(19,27,46),(20,28,47),(61,105,87),(62,106,88),(63,107,89),(64,108,90),(65,109,91),(66,110,92),(67,111,93),(68,112,94),(69,113,95),(70,114,96),(71,115,97),(72,116,98),(73,117,99),(74,118,100),(75,119,81),(76,120,82),(77,101,83),(78,102,84),(79,103,85),(80,104,86)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,66,11,76),(2,65,12,75),(3,64,13,74),(4,63,14,73),(5,62,15,72),(6,61,16,71),(7,80,17,70),(8,79,18,69),(9,78,19,68),(10,77,20,67),(21,118,31,108),(22,117,32,107),(23,116,33,106),(24,115,34,105),(25,114,35,104),(26,113,36,103),(27,112,37,102),(28,111,38,101),(29,110,39,120),(30,109,40,119),(41,99,51,89),(42,98,52,88),(43,97,53,87),(44,96,54,86),(45,95,55,85),(46,94,56,84),(47,93,57,83),(48,92,58,82),(49,91,59,81),(50,90,60,100)]])
C3×Dic10 is a maximal subgroup of
C20.D6 C15⋊SD16 C15⋊Q16 C3⋊Dic20 D12⋊D5 D60⋊C2 D15⋊Q8 C3×Q8×D5 Dic10.A4
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | D5 | D10 | C3×Q8 | C3×D5 | Dic10 | C6×D5 | C3×Dic10 |
| kernel | C3×Dic10 | C3×Dic5 | C60 | Dic10 | Dic5 | C20 | C15 | C12 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C3×Dic10 ►in GL2(𝔽19) generated by
| 7 | 0 |
| 0 | 7 |
| 8 | 11 |
| 12 | 0 |
| 11 | 4 |
| 17 | 8 |
G:=sub<GL(2,GF(19))| [7,0,0,7],[8,12,11,0],[11,17,4,8] >;
C3×Dic10 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{10} % in TeX
G:=Group("C3xDic10"); // GroupNames label
G:=SmallGroup(120,16);
// by ID
G=gap.SmallGroup(120,16);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-5,60,141,66,2404]);
// Polycyclic
G:=Group<a,b,c|a^3=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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